From Hilbert's programme to ordinal analysis.

Ordinal analysis of theories is a core area of proof
theory. The origins of proof theory can be traced back to the
second problem on Hilbert's famous list of problems. Proof theory
was invented as the main tool for carrying out Hilbert's
programme. In the main, ordinal-theoretic proof theory came into
existence in 1936, springing forth from Gentzen's consistency
proof of arithmetic. The intent of the talk is to elucidate the
underlying notions and the rationale of ordinal-theoretic proof
theory by relating the developments from Gentzen up to recent
advances in ordinal analysis of strong theories.